Turning to relative value, the recent volatility has presented opportunities at the front end of the curve. In particular, the 3.75% Apr-26s/ 4.625% Sep-26s curve appears 1.8bp too steep relative to the shape of the 2s/5s par curve (Figure 17). Hence, we recommend initiating 100:97 weighted 3.75% Apr-26/ 4.625% Sep-26 flatteners.
In [45]:
import sys
sys.path.append("../../")
In [46]:
from CurveInterpolator import GeneralCurveInterpolator
from CurveDataFetcher import CurveDataFetcher
from utils.rv_utils import cusip_spread_rv_regression
from utils.viz import plot_usts
from models.calibrate import calibrate_mles_ols, calibrate_nss_ols
from models.NelsonSiegelSvensson import NelsonSiegelSvenssonCurve
In [47]:
import pandas as pd
import numpy as np
import scipy
from datetime import datetime
from typing import Dict, List
import tqdm
import matplotlib.pyplot as plt
import matplotlib.pylab as pylab
params = {
"axes.titlesize": "x-large",
"legend.fontsize": "x-large",
"axes.labelsize": "x-large",
"xtick.labelsize": "x-large",
"ytick.labelsize": "x-large",
}
pylab.rcParams.update(params)
import seaborn as sns
sns.set(style="whitegrid", palette="dark")
import nest_asyncio
nest_asyncio.apply()
import warnings
warnings.filterwarnings('ignore', category=FutureWarning)
warnings.filterwarnings('ignore', category=pd.errors.SettingWithCopyWarning)
warnings.filterwarnings('ignore', category=RuntimeWarning)
import plotly
plotly.offline.init_notebook_mode()
%load_ext autoreload
%autoreload 2
The autoreload extension is already loaded. To reload it, use: %reload_ext autoreload
In [48]:
curve_data_fetcher = CurveDataFetcher(use_ust_issue_date=True)
In [49]:
quote_type = "eod"
as_of_date = datetime(2024, 4, 12)
curve_set_df = curve_data_fetcher.build_curve_set(
as_of_date=as_of_date,
sorted=True,
include_off_the_run_number=True,
market_cols_to_return=[f"{quote_type}_price", f"{quote_type}_yield"],
calc_free_float=True,
use_github=True,
)
curve_set_df
Out[49]:
| cusip | security_type | auction_date | issue_date | maturity_date | time_to_maturity | int_rate | high_investment_rate | is_on_the_run | ust_label | ... | parValue | percentOutstanding | est_outstanding_amt | corpus_cusip | outstanding_amt | portion_unstripped_amt | portion_stripped_amt | reconstituted_amt | free_float | rank | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 912797JN8 | Bill | 2024-03-14 | 2024-03-19 | 2024-04-16 | 0.010959 | NaN | 5.375 | False | 5.375% Apr-24 | ... | 8.004506e+08 | 0.003311 | 2.417805e+11 | NaN | 0.000000e+00 | NaN | 0.000000e+00 | NaN | -800.4506 | 16.0 |
| 1 | 912797JP3 | Bill | 2024-03-21 | 2024-03-26 | 2024-04-23 | 0.030137 | NaN | 5.365 | False | 5.365% Apr-24 | ... | 8.036739e+08 | 0.003467 | 2.317933e+11 | NaN | 0.000000e+00 | NaN | 0.000000e+00 | NaN | -803.6739 | 15.0 |
| 2 | 912797JQ1 | Bill | 2024-03-28 | 2024-04-02 | 2024-04-30 | 0.049315 | NaN | 5.381 | False | 5.381% Apr-24 | ... | 6.621273e+08 | 0.002987 | 2.216449e+11 | NaN | 0.000000e+00 | NaN | 0.000000e+00 | NaN | -662.1273 | 14.0 |
| 3 | 912797JV0 | Bill | 2024-04-04 | 2024-04-09 | 2024-05-07 | 0.068493 | NaN | 5.360 | True | 5.360% May-24 | ... | 6.495379e+08 | 0.002998 | 2.166564e+11 | NaN | 0.000000e+00 | NaN | 0.000000e+00 | NaN | -649.5379 | 13.0 |
| 4 | 912797JW8 | Bill | 2024-03-14 | 2024-03-19 | 2024-05-14 | 0.087671 | NaN | 5.393 | False | 5.393% May-24 | ... | 3.774967e+08 | 0.002579 | 1.463749e+11 | NaN | 0.000000e+00 | NaN | 0.000000e+00 | NaN | -377.4967 | 12.0 |
| ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... |
| 383 | 912810TN8 | Bond | 2023-04-13 | 2023-04-17 | 2053-02-15 | 28.865753 | 3.625 | NaN | False | 3.625% Feb-53 | ... | 9.367420e+09 | 0.141188 | 6.634703e+10 | 912803GP2 | 6.634703e+10 | 53099246.7 | 1.324778e+10 | 718095.5 | 43731.8268 | 4.0 |
| 384 | 912810TR9 | Bond | 2023-07-13 | 2023-07-17 | 2053-05-15 | 29.109589 | 3.625 | NaN | False | 3.625% May-53 | ... | 5.723292e+09 | 0.091264 | 6.271107e+10 | 912803GS6 | 6.271107e+10 | 42925380.8 | 1.978569e+10 | 584400.0 | 37202.0889 | 3.0 |
| 385 | 912810TT5 | Bond | 2023-10-12 | 2023-10-16 | 2053-08-15 | 29.361644 | 4.125 | NaN | False | 4.125% Aug-53 | ... | 8.606596e+09 | 0.120232 | 7.158330e+10 | 912803GU1 | 7.158330e+10 | 62556573.0 | 9.026728e+09 | 522400.0 | 53949.9768 | 2.0 |
| 386 | 912810TV0 | Bond | 2024-01-11 | 2024-01-16 | 2053-11-15 | 29.613699 | 4.750 | NaN | False | 4.750% Nov-53 | ... | 4.567153e+08 | 0.006874 | 6.644364e+10 | 912803GW7 | 6.644364e+10 | 58846008.9 | 7.597636e+09 | 270800.0 | 58389.2936 | 1.0 |
| 387 | 912810TX6 | Bond | 2024-03-13 | 2024-03-15 | 2054-02-15 | 29.865753 | 4.250 | NaN | True | 4.250% Feb-54 | ... | 1.848935e+09 | 0.037860 | 4.883608e+10 | 912803GY3 | 4.883608e+10 | 46311497.8 | 2.524586e+09 | 126500.0 | 44462.5632 | 0.0 |
388 rows × 25 columns
Build Par Curve Model¶
In [50]:
def liquidity_premium_curve_set_filter(curve_set_df: pd.DataFrame):
# remove OTRs, olds, double olds, triple olds
curve_set_filtered_df = curve_set_df[
(curve_set_df["rank"] != 0) & (curve_set_df["rank"] != 1) & (curve_set_df["rank"] != 2) & (curve_set_df["rank"] != 3)
]
# remove TBills
curve_set_filtered_df = curve_set_filtered_df[curve_set_filtered_df["security_type"] != "Bill"]
# remove low free float bonds (< $5bn)
curve_set_filtered_df = curve_set_filtered_df[curve_set_filtered_df["free_float"] > 5000]
# filter out bonds very close to maturity
curve_set_filtered_df = curve_set_filtered_df[curve_set_filtered_df["time_to_maturity"] > 30 / 360]
# remove CTDs
curve_set_filtered_df = curve_set_filtered_df[
~curve_set_filtered_df["cusip"].isin(
[
curve_data_fetcher.ust_data_fetcher.cme_ust_label_to_cusip("4.625s 2026-09-15")["cusip"], # TU
curve_data_fetcher.ust_data_fetcher.cme_ust_label_to_cusip("4.125s 2027-09-30")["cusip"], # Z3N
curve_data_fetcher.ust_data_fetcher.cme_ust_label_to_cusip("4.25s 2029-02-28")["cusip"], # FV
curve_data_fetcher.ust_data_fetcher.cme_ust_label_to_cusip("4.25s 2031-06-30")["cusip"], # TY
curve_data_fetcher.ust_data_fetcher.cme_ust_label_to_cusip("4.375s 2034-05-15")["cusip"], # TN
curve_data_fetcher.ust_data_fetcher.cme_ust_label_to_cusip("4.625s 2040-02-15")["cusip"], # US
curve_data_fetcher.ust_data_fetcher.cme_ust_label_to_cusip("4.5s 2044-02-15")["cusip"], # TWE
curve_data_fetcher.ust_data_fetcher.cme_ust_label_to_cusip("4.75s 2053-11-15")["cusip"], # UL
]
)
]
curve_set_filtered_df = curve_set_filtered_df.sort_values(by=["time_to_maturity"])
return curve_set_filtered_df
def no_filter(curve_set_df: pd.DataFrame):
return curve_set_df
In [51]:
# filter and fit bspline w/ knots are liquidity points
curve_set_filtered_df = liquidity_premium_curve_set_filter(curve_set_df=curve_set_df)
filtered_fitted_interpolator = GeneralCurveInterpolator(
x=curve_set_filtered_df["time_to_maturity"].to_numpy(),
y=curve_set_filtered_df[f"{quote_type}_yield"].to_numpy(),
)
fitted_bspline = filtered_fitted_interpolator.b_spline_with_knots_interpolation(
knots=[0.5, 1, 2, 3, 4, 5, 7, 10, 12, 15, 20, 25],
k=3,
return_func=True,
)
nss_func, status_nss, _ = calibrate_nss_ols(
curve_set_filtered_df["time_to_maturity"].to_numpy(),
curve_set_filtered_df[f"{quote_type}_yield"].to_numpy(),
)
assert status_nss
mles_func, status_mles = calibrate_mles_ols(
curve_set_filtered_df["time_to_maturity"].to_numpy(),
curve_set_filtered_df[f"{quote_type}_yield"].to_numpy(),
overnight_rate=5.31,
N=9,
)
In [52]:
plot_usts(
curve_set_df=curve_set_df,
ttm_col="time_to_maturity",
ytm_col=f"{quote_type}_yield",
hover_data=[
"issue_date",
"maturity_date",
"cusip",
"original_security_term",
"ust_label",
f"{quote_type}_price",
"free_float",
],
ust_labels_highlighter=[
("3.750% Apr-26", "red"), ("4.625% Sep-26", "blue"),
],
zero_curves=[(fitted_bspline, "BSpline k=3 - Zero Filtered Fit"), (nss_func, "Nelson Siegel Svensson"), (mles_func, "Merrill Lynch Exponential Spline")],
title=f"All USTs - using {f"{quote_type}_yield"} - as of {as_of_date}",
y_axis_range=[4.3, 5.7]
)
Fetching historical curve sets to regress 3.75% Apr-26s/ 4.625% Sep-26s vs our fitted model over time (2s/5s par curve)¶
In [53]:
start_date = datetime(2024, 1, 1)
end_date = datetime(2024, 4, 10)
curve_sets_dict_df, fitted_curves_dict = curve_data_fetcher.fetch_historical_curve_sets(
start_date=start_date,
end_date=end_date,
fetch_soma_holdings=True,
fetch_stripping_data=True,
calc_free_float=True,
fitted_curves=[
("LPF", f"{quote_type}_yield", liquidity_premium_curve_set_filter),
("NSS", f"{quote_type}_yield", no_filter, calibrate_nss_ols),
],
)
FETCHING CURVE SETS...: 100%|██████████| 92/92 [00:01<00:00, 62.10it/s] AGGREGATING CURVE SET DFs: 100%|██████████| 92/92 [00:03<00:00, 23.61it/s]
In [54]:
cusip_timeseries: Dict[str, List[Dict[str, str | float | int]]] = {}
fitted_cubic_spline_timeseries: Dict[datetime, scipy.interpolate] = {}
fitted_bspline_timeseries: Dict[datetime, scipy.interpolate] = {}
fitted_smooth_spline_timeseries: Dict[datetime, scipy.interpolate] = {}
nss_timeseries: Dict[datetime, NelsonSiegelSvenssonCurve] = {}
for dt in tqdm.tqdm(curve_sets_dict_df.keys(), desc="Main Loop"):
fitted_cubic_spline = fitted_curves_dict[dt]["LPF"].b_spline_with_knots_interpolation(
knots=[0.5, 1, 2, 3, 4, 5, 7, 10, 15, 20, 25],
k=3,
return_func=True,
)
fitted_bspline = fitted_curves_dict[dt]["LPF"].b_spline_with_knots_interpolation(
knots=[0.5, 1, 2, 3, 4, 5, 7, 10, 15, 20, 25], k=5, return_func=True
)
fitted_smooth_spline = fitted_curves_dict[dt]["LPF"].b_spline_with_knots_interpolation(
knots=[0.5, 1, 2, 3, 5, 7, 10, 20], k=4, return_func=True
)
curr_nss_model = fitted_curves_dict[dt]["NSS"]
fitted_cubic_spline_timeseries[dt] = fitted_cubic_spline
fitted_bspline_timeseries[dt] = fitted_bspline
fitted_smooth_spline_timeseries[dt] = fitted_smooth_spline
nss_timeseries[dt] = curr_nss_model
curr_curve_set_df = curve_sets_dict_df[dt]
curr_curve_set_df["lpf_cubic_spline_spread"] = curr_curve_set_df["eod_yield"] - fitted_cubic_spline(curr_curve_set_df["time_to_maturity"])
curr_curve_set_df["lpf_bspline_spread"] = curr_curve_set_df["eod_yield"] - fitted_bspline(curr_curve_set_df["time_to_maturity"])
curr_curve_set_df["lpf_smooth_spline_spread"] = curr_curve_set_df["eod_yield"] - fitted_smooth_spline(curr_curve_set_df["time_to_maturity"])
curr_curve_set_df["nf_nss_spread"] = curr_curve_set_df.apply(
lambda row: row["eod_yield"] - curr_nss_model(row["time_to_maturity"])
if pd.notna(row["eod_yield"]) and curr_nss_model(row["time_to_maturity"]) is not None
else np.nan, axis=1
)
for _, row in curr_curve_set_df.iterrows():
if row["cusip"] not in cusip_timeseries:
cusip_timeseries[row["cusip"]] = []
payload = {
"Date": dt,
"cusip": row["cusip"],
f"{quote_type}_yield": row[f"{quote_type}_yield"],
f"{quote_type}_price": row[f"{quote_type}_price"],
"lpf_cubic_spline_spread": row["lpf_cubic_spline_spread"],
"lpf_bspline_spread": row["lpf_bspline_spread"],
"lpf_smooth_spline_spread": row["lpf_smooth_spline_spread"],
"nf_nss_spread": row["nf_nss_spread"],
"free_float": row["free_float"],
"est_outstanding_amount": row["est_outstanding_amt"],
"soma_holdings": row["parValue"],
"soma_holdings_percent_outstanding": row["percentOutstanding"],
"stripped_amount": row["portion_stripped_amt"],
"reconstituted_amount": row["reconstituted_amt"],
"lpf_cubic_spline": fitted_cubic_spline,
"lpf_bspline": fitted_bspline,
"lpf_smooth_spline": fitted_smooth_spline,
"nf_nss": curr_nss_model,
}
cusip_timeseries[row["cusip"]].append(payload)
Main Loop: 100%|██████████| 70/70 [00:02<00:00, 30.79it/s]
In [55]:
ct_yields_df = curve_data_fetcher.fedinvest_data_fetcher.get_historical_ct_yields(start_date=start_date, end_date=end_date)
ct_yields_df
Out[55]:
| Date | CT2M | CT3M | CT6M | CT1 | CT2 | CT3 | CT5 | CT7 | CT10 | CT20 | CT30 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 2024-01-02 | NaN | 5.414579 | 5.266761 | 4.812030 | 4.332596 | 4.100865 | 3.930802 | 3.951168 | 3.943937 | 4.247731 | 4.083212 |
| 1 | 2024-01-03 | NaN | 5.403556 | 5.277076 | 4.822615 | 4.332576 | 4.077939 | 3.895921 | 3.914869 | 3.906323 | 4.209600 | 4.052949 |
| 2 | 2024-01-04 | NaN | 5.403156 | 5.250537 | 4.855018 | 4.382398 | 4.145585 | 3.972914 | 3.992895 | 3.988823 | 4.295010 | 4.137418 |
| 3 | 2024-01-05 | NaN | 5.391337 | 5.238747 | 4.854037 | 4.399235 | 4.178916 | 4.015292 | 4.040103 | 4.048999 | 4.363246 | 4.207898 |
| 4 | 2024-01-08 | NaN | 5.390940 | 5.227726 | 4.831904 | 4.365890 | 4.121773 | 3.966166 | 3.993101 | 4.010992 | 4.322075 | 4.176737 |
| ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... |
| 65 | 2024-04-04 | NaN | 5.373238 | 5.348941 | 5.022444 | 4.649279 | 4.465782 | 4.300537 | 4.307803 | 4.308815 | 4.574195 | 4.468574 |
| 66 | 2024-04-05 | NaN | 5.382595 | 5.369280 | 5.064586 | 4.733240 | 4.546828 | 4.378591 | 4.392378 | 4.388936 | 4.647449 | 4.541568 |
| 67 | 2024-04-08 | NaN | 5.382196 | 5.379642 | 5.085891 | 4.783829 | 4.604838 | 4.428457 | 4.429570 | 4.417144 | 4.654829 | 4.549541 |
| 68 | 2024-04-09 | NaN | 5.312104 | 5.326863 | 5.152263 | 4.935888 | 4.791008 | 4.607524 | 4.600711 | 4.563138 | 4.776680 | 4.650411 |
| 69 | 2024-04-10 | NaN | 5.415888 | 5.421877 | 5.204843 | 4.970175 | 4.803073 | 4.607708 | 4.590076 | 4.555092 | 4.759129 | 4.634122 |
70 rows × 12 columns
In [56]:
label1 = "3.750% Apr-26"
label2 = "4.625% Sep-26"
cusip_spread_rv_regression(
curve_data_fetcher=curve_data_fetcher,
label1=label1,
label2=label2,
cusip_timeseries=cusip_timeseries,
fitted_splines_timeseries_dict={
"lpf_cubic_spline": fitted_cubic_spline_timeseries,
"lpf_bspline": fitted_bspline_timeseries,
"lpf_smooth_spline": fitted_smooth_spline_timeseries,
"lpf_smooth_spline": fitted_smooth_spline_timeseries,
},
benchmark_tenor_1=2,
benchmark_tenor_2=5,
ct_yields_df=ct_yields_df,
)
lpf_cubic_spline is Benchmark Spline
OLS Regression Results
=========================================================================================
Dep. Variable: 3.750% Apr-26 / 4.625% Sep-26 R-squared: 0.537
Model: OLS Adj. R-squared: 0.531
Method: Least Squares F-statistic: 78.98
Date: Sun, 06 Oct 2024 Prob (F-statistic): 5.40e-13
Time: 19:03:49 Log-Likelihood: 225.27
No. Observations: 70 AIC: -446.5
Df Residuals: 68 BIC: -442.0
Df Model: 1
Covariance Type: nonrobust
=========================================================================================
coef std err t P>|t| [0.025 0.975]
-----------------------------------------------------------------------------------------
const 0.0013 0.011 0.121 0.904 -0.020 0.023
lpf_cubic_spline_2s5s 0.2726 0.031 8.887 0.000 0.211 0.334
==============================================================================
Omnibus: 8.506 Durbin-Watson: 0.890
Prob(Omnibus): 0.014 Jarque-Bera (JB): 8.790
Skew: -0.616 Prob(JB): 0.0123
Kurtosis: 4.224 Cond. No. 29.4
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
OLS Regression Results
=========================================================================================
Dep. Variable: 3.750% Apr-26 / 4.625% Sep-26 R-squared: 0.549
Model: OLS Adj. R-squared: 0.542
Method: Least Squares F-statistic: 82.67
Date: Sun, 06 Oct 2024 Prob (F-statistic): 2.30e-13
Time: 19:03:50 Log-Likelihood: 226.13
No. Observations: 70 AIC: -448.3
Df Residuals: 68 BIC: -443.8
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const 0.0100 0.012 0.863 0.391 -0.013 0.033
CT2-CT5 0.2797 0.031 9.092 0.000 0.218 0.341
==============================================================================
Omnibus: 7.876 Durbin-Watson: 0.851
Prob(Omnibus): 0.019 Jarque-Bera (JB): 7.257
Skew: -0.684 Prob(JB): 0.0266
Kurtosis: 3.787 Cond. No. 30.2
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
Using lpf_cubic_spline for UST Metrics Calcs
3.750% Apr-26 Metrics Calc: 70it [00:02, 30.85it/s] 4.625% Sep-26 Metrics Calc: 70it [00:01, 37.84it/s]
